Traveling waves for the nonlocal Fisher Equation can exhibit much more complex behaviour
than for the usual Fisher equation. A striking numerical observation is that a traveling
wave with minimal speed can connect a dynamically unstable steady state 0 to a Turing
unstable steady state 1, see . This is proved
in [1, 6] in
the case where the speed is far from minimal, where we expect the wave to be monotone.
Here we introduce a simplified nonlocal Fisher equation for which we can build simple
analytical traveling wave solutions that exhibit various behaviours. These traveling
waves, with minimal speed or not, can (i) connect monotonically 0 and 1, (ii) connect
these two states non-monotonically, and (iii) connect 0 to a wavetrain around 1. The
latter exist in a regime where time dynamics converges to another object observed in
[3, 8]: a
wave that connects 0 to a pulsating wave around 1.